Coronavirus peplomer interaction

By virtue of their lack of motility, viruses rely entirely on their own temperature (Brownian motion) to position themselves properly for cell attachment. Spiked viruses use one or more spikes (called peplomers) to attach. The coronavirus uses adjacent peplomer pairs. These peplomers, identically charged, repel one another over the surface of their convex capsids to form beautiful polyhedra. We identify the edges of these polyhedra with the most important peplomer hydrodynamic interactions. These convex capsids may or may not be spherical, and their peplomer population declines with infection time. These peplomers are short, equidimensional, and bulbous with triangular bulbs. In this short paper, we explore the interactions between nearby peplomer bulbs. By interactions, we mean the hydrodynamic interferences between the velocity profiles caused by the drag of the suspending fluid when the virus rotates. We find that these peplomer hydrodynamic interactions raise rotational diffusivity of the virus, and thus affect its ability to infect.


I. INTRODUCTION
By virtue of their lack of motility, viruses rely entirely on their own temperature (Brownian motion) to position themselves properly for cell attachment. 1 For such attachment, the virus faces the Goldilocks problem: (i) too little rotational diffusivity and the alignment time will exceed the chemical attachment kinetics requirement and (ii) too much rotational diffusivity and the alignment time will subceed this kinetics requirement. 2,3 Spiked viruses use one or more spikes (called peplomers) to attach (Sec. I of Ref. 1). The coronavirus uses adjacent peplomer pairs. These peplomers, identically charged, repel one another over the surface of their convex capsids to form beautiful polyhedra. 4,5 These convex capsids may or may not be spherical, 6 and their peplomer population declines after infection. 7 These peplomers are short, equidimensional, and bulbous with triangular bulbs. 8 In this short paper, we approximate the spike hydrodynamics with those of a single sphere (see Sec. VII of Ref. 1).
To deepen our understanding of viruses, or the suspension of any particles of complex shape for that matter, we can replace the particle with a rigid bead-rod structure. The beads represent sites of local drag following Stokes flow. The (nonexistent) dimensionless and massless rods represent the rigidly fixed distances between adjacent or nearby beads. In this concise paper, we consider only the hydrodynamic interferences between nearby beads. By nearby, we mean beads whose centers terminate a common edge of the polyhedral solution to the Thomson problem (see this justified on p. 113101-2 of Sec. I in Ref. 13). We thus consider only spherical capsids, leaving aspherical ones for another day.
In this short paper, we explore the interactions between nearby peplomer bulbs. By interactions, we mean the hydrodynamic interferences between the velocity profiles caused by the drag of the suspending fluid when the virus rotates. We focus specifically on the average number of peplomers on a coronavirus particle immediately after infection, 4,9,10 that is, N p ¼ 74 (see Fig. 5 of Ref. 1). Tables I (with Fig. 1) and II define our symbols and variables, respectively, dimensional and dimensionless. Though our work is mainly driven by curiosity, its public health implications have not escaped our attention. For instance, we know that the relevant mutations differentiating successive variants of SARS-CoV-2 are mutations affecting the peplomer proteins. 11 How these peplomer mutations affect their shapes, sizes, or populations is however not known.
For general rigid bead-rod theory with hydrodynamic interaction, for the real and minus the imaginary parts of the coronavirus contribution to the complex viscosity, we get [Eqs. (23) and (24) of where the tildes signify with hydrodynamic interaction, namely, where A > 0. 10,12 The best way to getk is to fit Eq. (1) which we will use below. We find that for N p ¼ 74, 12 peplomers form just five polyhedral edges, and 62 form six edges. We count 216 edges on our 74-vertex peplomer polyhedron, which Fig. 2 (Multimedia view) illustrates and animates. We further find that these nearby peplomer hydrodynamic interactions raise the rotational diffusivity of the virus, and thus affect its ability to infect.
General rigid bead-rod theory distinguishes itself from theories for a suspension of spheres, about which general rigid bead-rod theory is silent. General rigid bead-rod theory, after all, arrives at the complex viscosity through the orientation distribution function in small-

Name
Unit Symbol M=bead m Bead friction coefficient M=t f Bead radius (Fig. 1) L r b d=2 Capsid radius (Fig. 1) L r c Characteristic length (Fig. 1) L L Energy values in molecular-scale systems Minus imaginary part of complex viscosity (modified) M=Ltg 00 Moments of inertia (modified) ML 2Ĩ 1 ;Ĩ 2 ;Ĩ 3 Peplomer bulb radius L r p Position vector of the ith bead and jth element with respect to the center of mass

L R ij
Radius difference, capsid minus bead ( Fig. 1) Real part of complex viscosity (modified)

Hydrodynamic interaction parameter [Eq. (6)]
A d=2L Modified coefficients in the expression for the shear relaxation function in columns 5-7 of Table IVã ;b; Modified lopsidedness in column 8 of Table IV 2b a Total number of beads N Total number of peplomers N p Total number of capsid beads N c Characteristic bead-rod length ' L=x c Physics of Fluids ARTICLE scitation.org/journal/phf amplitude oscillatory shear flow. 13 By contrast, spheres in suspension are without orientation.

II. METHOD
In this paper, we employ the recent method of Secs. II of Refs. 8 and 14, and specifically, we neglect "other" terms in Eqs. (29)-(30) and (18)- (19), respectively. By other, we mean peplomer hydrodynamic interactions other than those between ends of each polyhedral edge. Figure 3 illustrates the structure of the peplomer population N p ¼ 74 spread over the surface of the capsid. We find 12 peplomers (blue in Fig. 3) with five nearby neighbors, and 62 peplomers (red in Fig. 3) with six. In other words, the peplomer population arranges itself onto 74 vertices of the 144-sided polyhedron, with 216 edges. Our Thomson solutions of course satisfy Euler's theorem, We tabulate, in Cartesian coordinates, the peplomer positions for N p ¼ 74. Peplomer positions marked in blue in Table III have just five nearby neighbors, and the rest, six. On average, each peplomer has 216/37 nearby neighbors.
From Table II, we learn that A d=2L, namely, that the extent of the hydrodynamic interaction between peplomers is dimensionless, and just depends on peplomer geometry. Specifically, it depends upon peplomer size and separation. Combining the dimensionless length and and l ¼ 0:5529. The physical dimensions of peplomers thus dictate the following range for peplomer hydrodynamic interaction: 9:28 100A 10:96: Therefore, throughout this work we parametrize peplomer hydrodynamic interaction as A ¼ 0:09; 0:10; 0:11. The L entering Eq. (6) is the center-to-center distance between nearest neighbors. This separation will of course differ from one peplomer population, N p , to another. As a consequence, the range Eq. (8) may or may not apply when N p differs significantly from 74. This short paper just focuses on N p ¼ 74, namely, the average peplomer population immediately after infection. In previous work, which neglects hydrodynamic interaction, we set d ¼ L arbitrarily. Otherwise put, so long as hydrodynamic interaction is neglected we can osculate beads (see Table IX of Ref. 1). However, when incorporating hydrodynamic interaction, we must set L ¼ 0:5529x c (l ¼ 0:5529), as we have done herein.

III. RESULTS
From Fig. 4, we learn that over the physical range of the hydrodynamic interaction parameter, Eq. (8), the real part of the complex viscosity of the coronavirus suspension decreases with hydrodynamic interaction. Otherwise put, hydrodynamic interaction makes the coronavirus suspension more non-Newtonian. However, from Fig. 4 we also learn that the real part of the complex viscosity decreases only slightly with frequency (witness the ordinate gradation magnitude). From Fig. 5, we learn that over the range Eq. (8), minus the imaginary part of the complex viscosity of the coronavirus suspension increases with hydrodynamic interaction. Otherwise put, hydrodynamic interaction makes the coronavirus suspension more non-Newtonian in this sense too. However, from Fig. 5, we also learn that minus the

ARTICLE
scitation.org/journal/phf imaginary part of the complex viscosity increases only slightly with frequency (witness the ordinate gradation magnitude). By non-Newtonian, we thus mean either the rise of minus the imaginary part or the fall of the real part of the complex viscosity.
which falls well above the value without hydrodynamic interaction, k 0Dr ¼ 3:36 Â 10 4 . Otherwise put, we find that hydrodynamic interaction of peplomers of the fully populated capsid increases its rotational diffusivity, and as the peplomer bulbs enlarge, the rotational diffusivity increases. Equation (9) and its companion Eq. (8) are the main results of this work.

IV. CONCLUSION
In this short paper, we explore the interactions between nearby peplomer bulbs (Sec. II). By interactions, we mean the hydrodynamic interferences between the velocity profiles caused by the drag of the suspending fluid around each peplomer when the virus rotates [defined by Eq. (6)]. We identify the primary hydrodynamic interactions with the edges of the 74-vertex polyhedron that is the solution to the Thomson problem (illustrated and animated in Fig. 2). We find that, for the wellknown dimensions of the coronavirus [Eq. (7)], we get the physical range for the coronavirus hydrodynamic interactions. Said physical range, Eq. (8), is a main result of this work. We further find that coronavirus peplomer hydrodynamic interactions raise rotational diffusivity of the virus, and thus affect its ability to infect (column 11 of Table IV). We also find that over Eq. (8), the physical range of the hydrodynamic interaction parameter, A, the real part of the complex viscosity of the coronavirus suspension decreases with hydrodynamic interaction (Fig. 4). Otherwise put, hydrodynamic interaction makes the coronavirus suspension more non-Newtonian. Finally, we also learn that over the range Eq. (8), minus the imaginary part of the complex viscosity of the coronavirus, suspension increases with hydrodynamic interaction (Fig. 5). Otherwise put, hydrodynamic interaction makes the coronavirus suspension more non-Newtonian in this sense too.
In this work, we focus entirely on N p ¼ 74, that is, on the fully populated virus at the time of infection. Our previous work on coronavirus rotational diffusivity without peplomer hydrodynamic interaction, k 0 D r , spans the range 10 N p 100 (see abscissa of  Table III) for each N p . This daunting task will require automation, and we leave it for another day.
We have restricted this work to spherical capsids, but we know coronavirus to be pleomorphic, 4 and we know that this pleomorphism matters to its rotational diffusivity (see column 11 of Table III  We have further restricted this work to the lowest energy peplomer arrangement, 19,20 that is, the most likely polyhedral solution to the Thomson problem; however, we know of less likely solutions, and thus expect the N p ¼ 74 population to be a dispersity of polyhedral arrangements. We leave the exploration of this dispersity (and the corresponding incorporation of its hydrodynamic interactions) for another day.

DATA AVAILABILITY
The data that support the findings of this study are available within the article.